@Varzar : i like your honesty. Thanks for bringing back this important topic.
For your first example about the death of princess diana, from my observation, i think you're right that you lack empathy in that situation.
@xudo too, the point is, it's NOT about whether YOU yourself care about princess diana or not. The event was somehow impactful to your family; means it's something important for your family. But, you dissed their feeling; you dissed the importance of the event to them. Means, you didn't respect their feelings (or opinion). I do understand what you said to them is very logical, that you don't understand why they need to feel sad; but in the same time, you implied to them that they are stupid for being sad. If the same thing being said about what we respect or feel important about, we would feel angry or uncomfortable too.
And for your second example, yes i think you might lack empathy (although better than the first one) because of the lack ability to console the said person. I think i do understand what you said about that laughing or about dont know how to console that person.
This relates too with what
@Kalinychta posted about empathy. From the example given in that post too, it could also be said that the lack of empathy is true, because of the lack of action to support the people in regards of whatever 'empathy' you might be feeling.
I think it can be said that "lack of empathy" means "lack of the ability to be reliable to the people who need your (emotional?) support when he/she needs it." The lack of ability to support other people when they need you. If you support the person with your actions rather than word (like fixing something for him/her), it's better than ignoring the person.
Another term is "the lack to be genuinely gentleman from the heart". (For women, please find similar word to gentleman.) A usual example is, a man saw his wife fell, but he didnt help her get up at all. The man said it's because he confirms that she didnt get hurt and she can get up by herself without his help. That's true. But at least a gentleman will help his lady to get up. Aspies can do this while masking. If during non-masking, maybe depends on person and situation (every aspie is different, maybe).
And it's maybe due to alexythimia, like
@Autistamatic said. Or maybe it's due to an aspie/autie can only manage to think about him/herself, always in the mindset of getting support, instead of thinking/trying to support other people.
Maybe this post is not welcomed, but i think it's important to be aware of it or discuss it. I might be wrong too.